Bias dependent tunneling in ferromagnetic junctions and inversion of the tunneling magnetoresistance from a quantum mechanical point of view

In the framework of the free-electron approximation, we have developed a quantum mechanical treatment for describing the bias dependent tunneling in FM/I/FM ferromagnetic junctions. In our theory, the Slonczewski model is extended to include the bias effect. In the barrier region, the Wentzel–Kramers–Brillouin wave function is used following Harrison. The main point of our treatment is to match the wave functions at both sides of the electrode/barrier interfaces quantum mechanically. We find that apart from the usual density of states effect, there exists a quantum coherent factor ${\mathrm{D(E}}_x{\mathrm{,V)=\kappa }}_R^2{\mathrm{(E}}_x{\mathrm{,V)-k}}_{\mathrm{R\uparrow }}{\mathrm{(E}}_x{\mathrm{,V)k}}_{\mathrm{R\downarrow }}{\mathrm{(E}}_x\mathrm{,V),}$ which decreases monotonously with the increasing applied bias and could change its sign at a sufficiently high bias. The characteristic of this coherent factor can explain the observed rapid decrease of tunneling magnetoresistance (TMR) with increasing bias and the sign change of TMR in some ferromagnetic junctions. Furthermore, numerical results for asymmetry barrier junctions provide a good understanding to the observed asymmetry of the TMR versus $V$ curve in junctions with composite barrier (e.g., ${\mathrm{Al}}_2{\mathrm{O}}_3/{\mathrm{Ta}}_2{\mathrm{O}}_5\mathrm{).}$